8 ms·

Geometry, Algebra, and Intuition (2017)

244 points by sebg 7y ago

georgewsinger 7y ago
Excellent post. In my experience, algebraic thinkers absorb information much faster than visual thinkers, but they more often make silly conceptual errors that visual thinkers don't make. For example, an algebraic thinker might accidentally add a vector to a scalar, since their symbols look identical on paper. But a visual thinker would be much less likely to do this, since their visual representations for scalars and vectors would likely be so distinct. A rule of thumb: when short-term speed is crucial, think algebraically. When long-term understanding is crucial, think visually. One problem with algebraic intuition is that it leave ideas "unhooked" in your mind. I mean this in the following sense: > While you are leaning things you need to think about them and examine them from many sides. By connecting them in many ways with what you already know.... you can later retrieve them in unusual situations. It took me a long time to realize that each time I learned something I should put "hooks" on it. This is another face of the extra effort, the studying more deeply, the going the extra mile, that seems to be characteristic of great scientists. -- Richard Hamming Algebraic proofs are stored as symbolic/syntactic movies in your head. But syntactic movies resemble other syntactic movies, causing algebraic proofs to blend together with all of the other symbolic/syntactic theorems. Visualizing proofs, on the other hand, makes each theorem significantly more distinguished from each other. You are much more likely to recall and understand important facts this way, in my opinion. You are therefore more likely to apply them in novel ways to solve new problems. Here Einstein famously describes visual vs. syntactic thinking in a letter to Jacques S. Hadamard: > (A) The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced and combined. > There is, of course, a certain connection between those elements and relevant logical concepts. It is also clear that the desire to arrive finally at logically connected concepts is the emotional basis of this rather vague play with the above-mentioned elements. But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought — before there is any connection with logical construction in words or other kinds of signs which can be communicated to others. > (B) The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will. > (C) According to what has been said, the play with the mentioned elements is aimed to be analogous to certain logical connections one is searching for. > (D) Visual and motor. In a stage when words intervene at all, they are, in my case, purely auditive, but they interfere only in a secondary stage, as already mentioned. A good friend of mine has a PhD in physics, and is a classic algebraic thinker. He is so, so much faster than me. But he once remarked that he forgets the proofs of almost everything he learned in grad school, and has to get back into the concrete exercises to regain his algebraic intuition. Visual thinkers may be slow, but they never forget.
- mlyle 7y ago
  IMO the post is a little circular. If we rely upon the projection product of Euclidean vectors, we've already granted Pythagorean theorem in our assumptions. There's a lot of ways to arrange things visually, but knowing we want c^2 really cuts the options down; knowing that we also will have a and b be degree two in the relation pretty much constrains us to that shape. We need a and b in some form on the sides, and we need a square with sides c. If we prefer, once we have the "4 triangles" model, it's easy for us to proceed to elementary algebra if we want, rather than relying on a geometrical transformation: (a+b)(a+b) area of the big square 1/2 (ab) area of each of the triangles (a+b)(a+b) - 4 * 1/2 (ab) = c^2 take the area of the big square, take the little triangles out, only the c^2 square remains a^2 + 2ab + b^2 - 4 * 1/2 ab = c^2 distribute a^2 + b^2 = c^2 simplify If you hate 4 triangles, you can easily do it with 2 of the a by b triangles, and a c by c right triangle forming a trapezoid. There's myriad geometrical constructions to start with before we get to the algebra. But we need to have some kind of geometric construction that leads to the algebra to conclude geometrical relations from algebraic relations.
  - jacobolus 7y ago
    > If we rely upon the projection product of Euclidean vectors, we've already granted Pythagorean theorem in our assumptions. The Pythagorean theorem in particular (and any theorem in general) is always a little bit “circular”; the relation is inherent in any definition of perpendicular in a model of Euclidean space. In geometric algebra (where multiplication of vectors distributes over addition), the two statements a² + b² = (a + b)² ⇔ ab + ba = 0 are obviously equivalent, so taking either of them as a definition for “perpendicular” immediately proves the other.
    - mlyle 7y ago
      Eh, I guess any proof is circular, but it is, IMO, "more" circular when the thing we're trying to prove was one of the preconditions of how we defined the system--- properties that we wanted to obtain a priori. It's like defining negative exponents based on the properties we want for commutation, identity, etc... and then afterwards proving that something raised to a negative exponent times something raised to a positive exponent is 1 and patting ourselves on the back.
adamnemecek 7y ago
Good thing Geometric Algebra exists, where the difference between the two is really blurred. Check out bivector.net, a new community for GA enthusiasts. Join us on discord https://discord.gg/vGY6pPk https://discord.gg/vGY6pPk The revolution is nigh. Check out this demo of ganja.js, a GA JS framework affiliated with bivector https://observablehq.com/@enkimute/animated-orbits https://observablehq.com/@enkimute/animated-orbits
- eggy 7y ago
  I find myself needing to move between both to really understand something, however, once I understand the problem I am working on I love using geometric algebra to play with it. I love Pablo Colapinto's Versor library and most of his writings and work [1]. [1] http://versor.mat.ucsb.edu/ http://versor.mat.ucsb.edu/
- jwmerrill 7y ago
  You might like some of the other posts on the blog! I’ve written several other posts showing how Geometric Algebra can be used in plane geometry problems that would typically be treated with lengths and angles. Most recent post, which links to other parts of the series: https://www.shapeoperator.com/2019/02/10/proving-theorems-about-angles-without-angles/ https://www.shapeoperator.com/2019/02/10/proving-theorems-ab... An older post applying GA to the geometry of sunsets: https://www.shapeoperator.com/2016/12/12/sunset-geometry/ https://www.shapeoperator.com/2016/12/12/sunset-geometry/
  - yiyus 7y ago
    Your sunset post was my first introduction to GA. What a rabbit hole! It has been one of the most fascinating subjects I have got into in the last years, and very useful for my work. Thank you!
- deleted 7y ago
  [deleted]
formalsystem 7y ago
If you enjoyed this post, you'll probably love "The Elements of Euclid"[1] by Byrne which provides entirely visual proof for ALL the basic proofs of euclidean geometry. I actually first came across the book when I saw it mentioned in Beautiful Explanations by Tufte. The beauty of the images is just on another level, the book will just make you feel good when you stare at it and after staring at it you'll absorb a proof accidentally with barely any effort on your part. There is a mistaken belief that visual proofs are less serious than algebraic ones but I believe this is mostly due to a lack of imagination when it comes coming up with good visual proofs. Byrne's book will help you see just how powerful pictures can be. There's lots of good work happening in the Category Theory community to turn diagrams into first class objects in constructing proofs so I'm very optimistic about a boom in visual proof construction. [1] https://www.amazon.com/Byrne-Six-Books-Euclid-Multilingual/dp/3836559382/ref=sr_1_3?keywords=euclid+elements&qid=1571798232&sr=8-3 https://www.amazon.com/Byrne-Six-Books-Euclid-Multilingual/d...
- xvilka 7y ago
  Could you please provide some links to these efforts for turning diagrams into proofs, please?
  - formalsystem 7y ago
    1. This was one of my favorite intro to category theory books, will make you familiar with the general language http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf 2. After skimming 1, check out Algebra Chapter 0 which uses Category Theory and diagrams to prove various theorems from Abstract Algebra (Hard read) https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813 https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Math...
- kqr 7y ago
  The Taschen reprint of Byrne's Euclid I consider one of the finer books I have. I have spent so much time with it. But there is now also this web version which is made with so much love it in many ways even improves on the reprint in quality: https://www.c82.net/euclid/ https://www.c82.net/euclid/
mindgam3 7y ago
This is great. Makes me want to do a similar post for chess. Whenever I try to explain to people what chess thinking involves I compare it geometry. Finding visual patterns on the board. But there’s also calculation and tactics which is very similar to algebraic thinking in terms of how you can derive a solution by following rules.
- dmix 7y ago
  I tried reading chess strategy books but it was always the mobile puzzle apps that really trained me to be a better player. Like programming books which demand you actually type out examples (which IMO is really useful) the same is true for almost all learning. Especially for something like math. Khan Academy mixed short instructional videos with quick tests which I found quite useful. but nothing beats thinking it out from scratch and building your own stuff.
avodonosov 7y ago
I'm suspicious of vector algebraic proof of the the Pythagorean Theorem. Don't vector operation properties themselves follow from the Pythagorean Theorem (at least in their application to space and geometric objects)? If so, using them to prove the theorem doesn't make sense. I'm not sure right now whether such circularity exists, but one should be careful.
- mkl 7y ago
  It seems circular to me. It relies on ||x||² = x·x, but the fact that ||x||² = x₁² + x₂² = x·x comes from the Pythagorean theorem.
  - posterboy 7y ago
    there are hundreds of proofs for the pythagorean theorem, surely some of them using the axioms of geometry
    - mkl 7y ago
      This isn't just using the axioms of geometry, though, it's trying to prove the Pythagorean theorem using the Pythagorean theorem as an axiom. Hence circularity.
      - posterboy 7y ago
        no, not exactly. the euclidean norm can be defined to fulfill a couple basic properties or equations (like composibility, or schwarz' inequality, but I don't recall exactly), and it's pure coincidence, if you will, that the norm is equal to the root mean square. That's not circular reasoning. The name euclidean norm implies that the geometric angle (no pun intended) was the motivation, but what's really central is a question of epistemology. There's not much of a point to prescribe a certain approach over another, without a good argument.
  - soVeryTired 7y ago
    I don't think it's quite circular, but in a sense you're right. if c = -a + b, then for any inner product you have <c,c> = <a,a> + <b,b> - 2<a,b> as a result of bi-linearity of the inner product. By definition if <a,b> = 0 then a and b are orthogonal and a version of pythagoras' theorem holds. But you're right in saying the choice of inner product <x,y> = x_1y_1 + ... + x_ny_n is motivated by pythagoras' theorem in Euclidean space.
29athrowaway 7y ago
https://abstrusegoose.com/376 https://abstrusegoose.com/376

244 points by sebg 7y ago

Excellent post. In my experience, algebraic thinkers absorb information much faster than visual thinkers, but they more often make silly conceptual errors that visual thinkers don't make. For example, an algebraic thinker might accidentally add a vector to a scalar, since their symbols look identical on paper. But a visual thinker would be much less likely to do this, since their visual representations for scalars and vectors would likely be so distinct. A rule of thumb: when short-term speed is crucial, think algebraically. When long-term understanding is crucial, think visually. One problem with algebraic intuition is that it leave ideas "unhooked" in your mind. I mean this in the following sense: > While you are leaning things you need to think about them and examine them from many sides. By connecting them in many ways with what you already know.... you can later retrieve them in unusual situations. It took me a long time to realize that each time I learned something I should put "hooks" on it. This is another face of the extra effort, the studying more deeply, the going the extra mile, that seems to be characteristic of great scientists. -- Richard Hamming Algebraic proofs are stored as symbolic/syntactic movies in your head. But syntactic movies resemble other syntactic movies, causing algebraic proofs to blend together with all of the other symbolic/syntactic theorems. Visualizing proofs, on the other hand, makes each theorem significantly more distinguished from each other. You are much more likely to recall and understand important facts this way, in my opinion. You are therefore more likely to apply them in novel ways to solve new problems. Here Einstein famously describes visual vs. syntactic thinking in a letter to Jacques S. Hadamard: > (A) The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced and combined. > There is, of course, a certain connection between those elements and relevant logical concepts. It is also clear that the desire to arrive finally at logically connected concepts is the emotional basis of this rather vague play with the above-mentioned elements. But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought — before there is any connection with logical construction in words or other kinds of signs which can be communicated to others. > (B) The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will. > (C) According to what has been said, the play with the mentioned elements is aimed to be analogous to certain logical connections one is searching for. > (D) Visual and motor. In a stage when words intervene at all, they are, in my case, purely auditive, but they interfere only in a secondary stage, as already mentioned. A good friend of mine has a PhD in physics, and is a classic algebraic thinker. He is so, so much faster than me. But he once remarked that he forgets the proofs of almost everything he learned in grad school, and has to get back into the concrete exercises to regain his algebraic intuition. Visual thinkers may be slow, but they never forget.

mlyle 7y ago
IMO the post is a little circular. If we rely upon the projection product of Euclidean vectors, we've already granted Pythagorean theorem in our assumptions. There's a lot of ways to arrange things visually, but knowing we want c^2 really cuts the options down; knowing that we also will have a and b be degree two in the relation pretty much constrains us to that shape. We need a and b in some form on the sides, and we need a square with sides c. If we prefer, once we have the "4 triangles" model, it's easy for us to proceed to elementary algebra if we want, rather than relying on a geometrical transformation: (a+b)(a+b) area of the big square 1/2 (ab) area of each of the triangles (a+b)(a+b) - 4 * 1/2 (ab) = c^2 take the area of the big square, take the little triangles out, only the c^2 square remains a^2 + 2ab + b^2 - 4 * 1/2 ab = c^2 distribute a^2 + b^2 = c^2 simplify If you hate 4 triangles, you can easily do it with 2 of the a by b triangles, and a c by c right triangle forming a trapezoid. There's myriad geometrical constructions to start with before we get to the algebra. But we need to have some kind of geometric construction that leads to the algebra to conclude geometrical relations from algebraic relations.
- jacobolus 7y ago
  > If we rely upon the projection product of Euclidean vectors, we've already granted Pythagorean theorem in our assumptions. The Pythagorean theorem in particular (and any theorem in general) is always a little bit “circular”; the relation is inherent in any definition of perpendicular in a model of Euclidean space. In geometric algebra (where multiplication of vectors distributes over addition), the two statements a² + b² = (a + b)² ⇔ ab + ba = 0 are obviously equivalent, so taking either of them as a definition for “perpendicular” immediately proves the other.
  - mlyle 7y ago
    Eh, I guess any proof is circular, but it is, IMO, "more" circular when the thing we're trying to prove was one of the preconditions of how we defined the system--- properties that we wanted to obtain a priori. It's like defining negative exponents based on the properties we want for commutation, identity, etc... and then afterwards proving that something raised to a negative exponent times something raised to a positive exponent is 1 and patting ourselves on the back.

adamnemecek 7y ago

Good thing Geometric Algebra exists, where the difference between the two is really blurred. Check out bivector.net, a new community for GA enthusiasts. Join us on discord https://discord.gg/vGY6pPk https://discord.gg/vGY6pPk The revolution is nigh. Check out this demo of ganja.js, a GA JS framework affiliated with bivector https://observablehq.com/@enkimute/animated-orbits https://observablehq.com/@enkimute/animated-orbits

eggy 7y ago
I find myself needing to move between both to really understand something, however, once I understand the problem I am working on I love using geometric algebra to play with it. I love Pablo Colapinto's Versor library and most of his writings and work [1]. [1] http://versor.mat.ucsb.edu/ http://versor.mat.ucsb.edu/
jwmerrill 7y ago
You might like some of the other posts on the blog! I’ve written several other posts showing how Geometric Algebra can be used in plane geometry problems that would typically be treated with lengths and angles. Most recent post, which links to other parts of the series: https://www.shapeoperator.com/2019/02/10/proving-theorems-about-angles-without-angles/ https://www.shapeoperator.com/2019/02/10/proving-theorems-ab... An older post applying GA to the geometry of sunsets: https://www.shapeoperator.com/2016/12/12/sunset-geometry/ https://www.shapeoperator.com/2016/12/12/sunset-geometry/
- yiyus 7y ago
  Your sunset post was my first introduction to GA. What a rabbit hole! It has been one of the most fascinating subjects I have got into in the last years, and very useful for my work. Thank you!
deleted 7y ago
[deleted]

formalsystem 7y ago

If you enjoyed this post, you'll probably love "The Elements of Euclid"[1] by Byrne which provides entirely visual proof for ALL the basic proofs of euclidean geometry. I actually first came across the book when I saw it mentioned in Beautiful Explanations by Tufte. The beauty of the images is just on another level, the book will just make you feel good when you stare at it and after staring at it you'll absorb a proof accidentally with barely any effort on your part. There is a mistaken belief that visual proofs are less serious than algebraic ones but I believe this is mostly due to a lack of imagination when it comes coming up with good visual proofs. Byrne's book will help you see just how powerful pictures can be. There's lots of good work happening in the Category Theory community to turn diagrams into first class objects in constructing proofs so I'm very optimistic about a boom in visual proof construction. [1] https://www.amazon.com/Byrne-Six-Books-Euclid-Multilingual/dp/3836559382/ref=sr_1_3?keywords=euclid+elements&qid=1571798232&sr=8-3 https://www.amazon.com/Byrne-Six-Books-Euclid-Multilingual/d...

xvilka 7y ago
Could you please provide some links to these efforts for turning diagrams into proofs, please?
- formalsystem 7y ago
  1. This was one of my favorite intro to category theory books, will make you familiar with the general language http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf 2. After skimming 1, check out Algebra Chapter 0 which uses Category Theory and diagrams to prove various theorems from Abstract Algebra (Hard read) https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813 https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Math...
kqr 7y ago
The Taschen reprint of Byrne's Euclid I consider one of the finer books I have. I have spent so much time with it. But there is now also this web version which is made with so much love it in many ways even improves on the reprint in quality: https://www.c82.net/euclid/ https://www.c82.net/euclid/

mindgam3 7y ago

This is great. Makes me want to do a similar post for chess. Whenever I try to explain to people what chess thinking involves I compare it geometry. Finding visual patterns on the board. But there’s also calculation and tactics which is very similar to algebraic thinking in terms of how you can derive a solution by following rules.

dmix 7y ago
I tried reading chess strategy books but it was always the mobile puzzle apps that really trained me to be a better player. Like programming books which demand you actually type out examples (which IMO is really useful) the same is true for almost all learning. Especially for something like math. Khan Academy mixed short instructional videos with quick tests which I found quite useful. but nothing beats thinking it out from scratch and building your own stuff.

avodonosov 7y ago

I'm suspicious of vector algebraic proof of the the Pythagorean Theorem. Don't vector operation properties themselves follow from the Pythagorean Theorem (at least in their application to space and geometric objects)? If so, using them to prove the theorem doesn't make sense. I'm not sure right now whether such circularity exists, but one should be careful.

mkl 7y ago
It seems circular to me. It relies on ||x||² = x·x, but the fact that ||x||² = x₁² + x₂² = x·x comes from the Pythagorean theorem.
- posterboy 7y ago
  there are hundreds of proofs for the pythagorean theorem, surely some of them using the axioms of geometry
  - mkl 7y ago
    This isn't just using the axioms of geometry, though, it's trying to prove the Pythagorean theorem using the Pythagorean theorem as an axiom. Hence circularity.
    - posterboy 7y ago
      no, not exactly. the euclidean norm can be defined to fulfill a couple basic properties or equations (like composibility, or schwarz' inequality, but I don't recall exactly), and it's pure coincidence, if you will, that the norm is equal to the root mean square. That's not circular reasoning. The name euclidean norm implies that the geometric angle (no pun intended) was the motivation, but what's really central is a question of epistemology. There's not much of a point to prescribe a certain approach over another, without a good argument.
- soVeryTired 7y ago
  I don't think it's quite circular, but in a sense you're right. if c = -a + b, then for any inner product you have <c,c> = <a,a> + <b,b> - 2<a,b> as a result of bi-linearity of the inner product. By definition if <a,b> = 0 then a and b are orthogonal and a version of pythagoras' theorem holds. But you're right in saying the choice of inner product <x,y> = x_1y_1 + ... + x_ny_n is motivated by pythagoras' theorem in Euclidean space.

29athrowaway 7y ago

https://abstrusegoose.com/376 https://abstrusegoose.com/376